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Herein we present a quantitative analysis of diffraction enhanced imaging (DEI) by evaluating object detectability. DEI is an X-ray imaging method that simultaneously produces absorption and refraction images of an object. In recent years, it has become widely accepted that task-based measures of image quality, such as the detectability of subtle image features, are the ultimate test of image quality. It is widely believed that the DEI refraction images measures soft tissue more sensitively than the DEI absorption image. In this paper, we use a modern task-based approach to evaluate this assertion. Specifically, we use the log-likelihood ration test to devise a statistically optimal detector, usually refer to as "ideal observer", and use the area under the receiver operating characteristic (ROC) curve as a detectability metric. Using this method, we study the detectability of a blood vessel in lung tissue at 18 keV. We find that the refraction image is better than the absorption image if the blood vessel is smaller than 0.4 cm, but that the absorption image is better for blood vessels larger than 0.4 cm. In future work, we will consider whether it is generally true that small objects are more detectable by refraction imaging while large objects are more detectable by absorption imaging.